I am currently in the settling process of the second batch of beer that I have ever brewed. It is one of those Pale Ale kits with the malt extract syrup, etc.

I brewed the wort on Saturday and began fermentation at around 5PM. The beginning gravity was 1.042 as the kit calls for. The airlock was bubbling really well as soon as I put the lid on, however, at around 7PM on Sunday there were no more bubbles. The kit states that it should ferment for 48-72 hours. So I popped the lid and took another gravity reading. I noticed that the yeast had floated to the top and was stagnant. My gravity reading was 1.020 at that point. Today, I took another reading and I can tell that the sediment was settling pretty well, but my gravity hasn't changed. However, the beer smelled alcoholic and smelled pretty good (no funky smells).

The kit states that my final gravity should be 1.010-1.012. Is the gravity going to get to that point after a few more days? Did it not ferment long enough and do I have to add more yeast and wait longer? Or is my beer going to be good to go and I'm overanalyzing a simple situation?

Here are some ideas that may help. If not this batch the next:
1. Always make a yeast starter a couple days ahead of brew day. This will ensure that you have plenty of good viable yeast ready for action.
2. Watch your fermentation temperature. Some yeast will stall if they get too cold.
3. Make sure that when you pitch the yeast into the wort that you aerate. (Shake the living crap out of it)

I think your FG is still too high. If you bottle your batch as is, my advice is to watch your carbonation level. After about a week of conditioning, sample a bottle. When you get where you want, get them in the frig or you might end up with exploding bottle.

... and don't really expect the fermentation to be done in a few days. It may happen, but the beer won't be finished anyway. Just let it sit for a week, at least. I wait from 2-3 weeks before I worry about transfering it, definately 3 weeks before bottling.

If you swirl it to resuspend the yeast a couple times a day, every day, for the rest of the week, you might get it to drop the gravity a bit more.

What do you mean by starter yeast? Or starting the yeast. I'm not too familiar with that. Also, Should I be letting the yeast sit in the wort 10 minutes before I stir? And should the yeast be evenly distributed throughout the batch? I guess where the root of my problems may be is at actually pitching the yeast, and the correct way to do it.

A yeast starter is a micro, approx 1 pint batch of low gravity, wort to get the yeast going prior to introducing it to the main batch. I usually use 1/3 cup DME, 2 cups water and two hop pellets. Boil for 5-10 min, cool and place in a flask with an air lock. When its aprox 80 degrees, I add the liquid yeast and shake shake shake to aerate. All this occurs 2 days before brew day so that the yeast has a chance to multiply and get busy. You can decant the liquid off the starter if you like but I usually shake and dump the starter into the main batch and then aerate by shaking the crap out of the carboy to get as much air into the wort as possible.
It sounds like you are using dry yeast. If that is the case, still follow the above for a successful fermentation. Stirring may mix the yeast in but shaking is a more effective way to aerate. I am not one that would try an aquarium pump and a stone to get more air into the wort. My theory is if it ain't broke, don't fix it.

I'm on my 3rd batch, I think were brewing the same thing. It's supposed to be an american pale ale. Started at 40 pts, after 3 days was only 20 pts. I did add another starter batch of yeast but it had little effect. Then. I just left it alone. Sure 'nuff, it got down to 12 pts.

It is not uncommon for extract brews to have a higher final gravity than planned. I had this happen with nearly every extract batch I made. The main idea is to get it down close -in the general area of expected FG. Time will usually do that for you, Patience grasshopper.
Since doing all grain batches I have noticed that my FG is almost always achieved. Im not real sure the exact science to this phenomena, but my brew notes consistently show that the same recipe done with extract yielded 3-4 points higher FG. To compenstae you could use a more attenuative yeast than the recipe calls for- or- try adding Beano- I do not recommend this procedure because it causes overattenuation and results in a higher alcohol beer than desired, but it does work.

It sounds like the beer just wasn't done yet. Perhaps it would have benefitted from swirling the yeast up a few times a day, after it settled, to keep it active.

I give it a week or two in primary before I worry about the gravity.

Also, don't make a starter if you use dry yeast. There should be plenty of yeast in a 12 gram pack of dry yeast for a typical beer, so there's no need to expand the number. Chances are your growth techniques aren't as good as the manufacturer's and you'll end up with yeast less healthy than what you had in the package. Do remember to aerate the wort well, as always.